Phase-insensitive unitary gate error
= Phase-insensitive unitary gate error
{title2=$J=1-|\operatorname{Tr}(U_T^\dagger U)|^2/n^2$}
The <Cauchy-Schwarz inequality> for the <Hilbert-Schmidt inner product> gives $0\leq J\leq1$ for $n$-dimensional <unitary operators>. Equality $J=0$ holds exactly when $U=e^{i\phi}U_T$. When absolute gate phase matters, use the phase-sensitive <Hilbert-Schmidt distance> $\|U-U_T\|^2/(2n)=1-\operatorname{Re}\operatorname{Tr}(U_T^\dagger U)/n$ instead.